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A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions [clarification needed]: homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate ...
It is common to place additional requirements on topological manifolds. In particular, many authors define them to be paracompact [3] or second-countable. [2] In the remainder of this article a manifold will mean a topological manifold. An n-manifold will mean a topological manifold such that every point has a neighborhood homeomorphic to R n.
A basic example of maps between manifolds are scalar-valued functions on a manifold, : or :, sometimes called regular functions or functionals, by analogy with algebraic geometry or linear algebra. These are of interest both in their own right, and to study the underlying manifold.
This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics . For categorical listings see Category:Manifolds and its subcategories.
An atlas for a topological space is an indexed family {(,):} of charts on which covers (that is, =).If for some fixed n, the image of each chart is an open subset of n-dimensional Euclidean space, then is said to be an n-dimensional manifold.
Local flatness is a property of a submanifold in a topological manifold of larger dimension. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds. Suppose a d dimensional manifold N is embedded into an n dimensional manifold M (where d < n).
A (topological) surface is a topological space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E 2. Such a neighborhood, together with the corresponding homeomorphism, is known as a (coordinate) chart. It is through this chart that the neighborhood inherits the standard coordinates on the ...
This category includes maps between manifolds, smooth or otherwise, particularly in geometric topology. Pages in category "Maps of manifolds" The following 14 pages are in this category, out of 14 total.